(t2.1) t2( p) =◦p, if p is an atomic formula;
(t2.2) t2(A #B) = t2(A) # t1(B), if # is any binary connective;
22 Might the reader observe that the first formulations of this result, on da Costa’s [49], Theorem 9, page 16, and on da Costa’s [50], Theorem 4, page 500, the general case in which an infinite number of atomic formulas occur in Γ ∪{A} is not considered.
(t2.3) t2(¬A) =¬t2(A);
(t2.4) t2(◦A) = ◦t2(A).
So, working with Cila, all we need to rely on in order to go on making ‘classi-cal inferences’ is on the consistency of the atomic constituents of our formulas. As a particular consequence of that, one can now substitute each new axiomatic rule of Cila by an alternative version in terms of ‘•’s instead of ‘◦’s. Thus, the axiom (ca3), for instance, can be rewritten as [•(A→B) (•A∨•B)] (use FACT 3.34(i) and COR-OLLARY 3.62). And so on.
Proposing an infinite number of calculi, instead of one, as in the case of the Cn, only starts to make sense after we prove that we are not just repeating the same tune:
THEOREM 3.63 Each Cn deductively extends each Cn+1, for 1≤n<ω.23
Proof: We will not here give it a try by usual ‘syntactical means’. For sure, this will be much easier to check if one just considers the semantics associated with these cal-culi, for instance in [53] (corrected in [74]) and in [36] (or [76]).
Evidently, all these calculi Cn extend also the calculus Cω —they even extend Cmin, the stronger logic on which we based bC, our first LFI (recall the subsection 3.1 for the definition of these logics). This Cω, we argue, was indeed a very bad choice as a kind of ‘limit’ to the hierarchy Cn, 1≤n<ω. Consider, for instance, the following result:
FACT 3.64 The only addition made by Cn (in fact, by Cia, for the axiom (cl) has no use in this result) to the rules provable by bC about the interdefinability of the bi-nary connectives (see THEOREM 3.18) is the rule (ix): [¬(A∧B) Cia (¬A∨¬B)], and its variants.
Proof: We just show that that rule holds already in Cia, and point the reader again to the semantical studies of the calculi Cn to check that the other formulas are still not prov-able in Cila. First of all, setting Γ= {◦(A∧B), ¬(A∧B), A} it can immediate be seen that [Γ, B Cia◦(A∧B)], [Γ, B Cia (A∧B)] and [Γ, B Cia¬(A∧B)], so we apply FACT 3.14(ii) to obtain [ΓCia¬B], and consequently [ΓCia (¬A∨¬B)]. But then, also [¬A Cia (¬A∨¬B)], and so the proof by cases will give us [◦(A∧B),
¬(A∧B) Cia (¬A∨¬B)]. By (ca1) we then conclude that [◦A, ◦B, ¬(A∧B) Cia (¬A∨¬B)], but we also have, from LEMMA 3.43(ii), that [Cia (¬A∨◦A)], and from that we obtain [◦B, ¬(A∧B) Cia (¬A∨¬B)]. By a similar reasoning, from [¬B Cia (¬A∨¬B)], we finally arrive at our goal, [¬(A∧B) Cia (¬A∨¬B)].
COROLLARY 3.65 (IpE) cannot hold in the calculi Cn, or in any extension of them.
Proof: Just recall THEOREM 3.51(iv).
The FACT 3.64 also suggests some further information about the plausibility of calling either Cω or Cmin ‘limits’ for the hierarchy Cn, 1≤n<ω. For, as we have seen in THEOREM 3.18, these new forms of De Morgan rules that we now have in each Cn
23 A supposedly general proof of this fact, dating still from the ‘syntactical period’, when no semantics had yet been presented to those calculi, appears for instance in da Costa’s [49], pp.17–9, and once again in Alves’s [2], pp.17–9, and is credited to Ayda Arruda. There is surely some mistake, however, in their attempt to prove the independence of each axiom [A(n), A, ¬A B] with respect to the axi-oms of Cn +1, given that this very axiom assumes non-distinguished values in all matrices Tn thereby presented, if one only picks 1 as the value of A and picks for B any value in between 1 and n +2.
are not present even in Cmin. Also, by a combination of FACT 3.40(i) and COROL-LARY 3.56, we know that ((A∧¬A)→¬¬(A∧¬A)) is valid in Cia, and so in each Cn, while we also know, from the matrices of P1, as in THEOREM 3.15(ii), that this formula cannot be a theorem of neither Cmin nor Cω. Now, it is only compelling to think of a deductive limit for an infinite hierarchy of increasingly weaker calculi as the logic having as inferences exactly all sets of inferences common to the whole hierarchy!24 As we have shown in [39], it is possible to define such a logic, for each hierarchy of dC-systems as the one given above, by way of the useful tool of possible-translations semantics, obtaining, as a byproduct, some clear-cut and effective decision procedures, even though some other very interesting questions, such as how to finitely axiomatize this limit-calculi, or how to define a strong negation in them, if this is pos-sible at all, were still left open (check also the paper [42]). Indeed, notice that when we go from each Cn to the following Cn+1 we need in fact to add a further require-ment in order to express the consistency of a formula A —while in Cn this was ex-pressible by way of A(n), or, equivalently, by way of the set {A1, A2, …, An}, in Cn+1
that same set must be incremented by the formula An+1. So, ultimately, in CLim, the de-ductive limit of the hierarchy Cn, the consistency of A can evidently be expressed by an infinite number of formulas, and again we obtain a logic which is gently explo-sive, being thus an LFI. What we still do not know is if logics such as CLim can be alternatively characterized in such a way so as to also reveal themselves as finitely gently explosive, like all the other logics presented up to this point, based on the axiom (bc1). If this characterization is not possible, it is hard to see how a strong negation or a bottom particle could then be defined in such a logic,25 so that this would make a
24 This logic Cω has puzzled people for too long as ‘part’ of the original hierarchy Cn. The existence of a logic as a real deductive limit to this hierarchy (see [39]) shows that it was clearly just a matter of coincidence that Cω would appear as a kind of ‘syntactical limit’ to the original axiomatic formulation of the Cn, by the deletion of all the axioms and definitions involving the connectives ◦ and • ((bc1), (ci) and (cl)), and the ‘deletion’ also of (Min9) (actually, this last axiom was not in the original formulat i-on of these calculi, and could not even be proved from the other axioms of Cω —see THEOREM 3.3).
To put the matter in clear terms, Cω can of course be studied in its own right, as a very weak paracon-sistent (non-LFI) logic based on positive intuitionistic logic, but it should not be seen as part of the hierarchy Cn, n≥1, for it has no more right to occupy that position than Cmin or many other logics that could substitute it would have!
This coincidence had also some harmful effects on the philosophical appreciation of the logics pro-duced by da Costa. As da Costa himself has put it in his original piece on these systems (cf. [49], p.21),
‘roughly speaking, we could say that human reason seems to attain the peak of its power the more it approaches the danger of trivialization’. This statement has been inspiring people to naively defend stances according to which, for instance, ‘the more a theory is useful to found mathematics, the more easily it results to trivialize it; and the more difficult it is to trivialize it, the less it is useful to found mathematics’ (see [26], p.243). There are good and bad points about these somewhat hasty conclusi-ons. First, as a general technical assertion about paraconsistent logics in general, da Costa’s motto is certainly misleading, given the existence of maximal logics such as the three-valued Pac (subsection 2.4), which is both as strong as a fragment of classical logic as it could be, and at the same time is not fini-tely trivializable at all. One could, then, restrict their attention to LFIs and repeat that motto in an environment in which it seems to make sense. In that case, of course, the second statement above would be affirming that no non-LFI could be useful to found mathematics —and this statement would be very likely to find its defensors (cf., for instance, Batens’s attack [13] on Priest’s [90]).
25 Here, we really mean defined inside the logic, as a real formula of this logic. For instance, in Priest’s [90] a logic containing no bottom particle is presented, but the author argues that such a propositional constant ⊥ could be ‘thought of informally as the conjunction of all formulas’ (p.146), so that, for instance, a strong negation ~ would be obtainable from that in the usual way, by letting ~A be defined
case in which the Gentle Principle of Explosion does not coincide with the Sup-plementing one, or with ex falso (compare this with FACT 2.19, and the comments which follow that result).
Some other interesting theses of Cila are the following (see Urbas’s [107]):
FACT 3.66 In Cila the schemas ◦⊥ and ◦ A are provable.
Proof: Recall first that A was the classical negation defined inside of bC (in THEOREM 3.48) as (A→⊥), and ⊥ is a bottom particle that can be defined, for in-stance, as (◦B∧(B∧¬B)), for some B. Now, by FACT 3.33 and COROLLARY 3.56, we know that both ◦◦B and ◦(B∧¬B) are theorems of Cil, and then we conclude, by the axiom (ca1), that ◦(◦B∧(B∧¬B)), and so ◦⊥ is a theorem of Cila. Recall also from FACT 3.49 that A is equivalent, in Ci, to ~A, and this last strong negation was defined as (¬A∧◦A), and so we have in particular that [(A→⊥) Cila◦A]. So, from this last inference and from the fact that ◦⊥ is a theorem of Cila, as proved above, we use (ca3) and conclude that [(A→⊥) Cila◦(A→⊥)]. As particular cases of the last two inferences, substituting A for ((A→⊥)→⊥) in the first case, and for (A→⊥) in the second case, we obtain, respectively, [ A Cila ◦ A] and [ A Cila◦ A], and the version of proof by cases obtained for leaves us at last with [Cila◦ A].
This last result gives us yet another reason for the failure of (IpE) in the Cn and in their extensions (COROLLARY 3.65) —now, it is THEOREM 3.51(i) that applies.
As we shall see in subsection 3.12, the failure of (IpE) makes it impossible for us to find a Lindenbaum-Tarski-like algebraization for these logics. In the case of the Cn the situation is actually worse: as Mortensen ([82]) has shown, no non-trivial congruence is definable for these logics, making these logics non-algebraizable even in a much more general sense, the one of Blok-Pigozzi (cf. THEOREM 3.83). There are, nev-ertheless, several extensions of the Cn in which non-trivial congruences can be de-fined, being thus much more receptive to algebraic treatments. We will be seeing many examples of these below.
Let us now investigate another way of propagating consistency, by liberalizing a little bit the conditions required by Cila (that is, C1). Da Costa, Béziau & Bueno pro-posed, in [57], to substitute the above axioms, (ca1)–(ca3), by the following:
(co1) (◦A∨◦B) ◦(A∧B);
(co2) (◦A∨◦B) ◦(A∨B);
(co3) (◦A∨◦B) ◦(A→B).
We will call Cilo the logic obtained by the addition of (co1)–(co3) to Cil. It is very easy to see, using the positive axioms, that this logic, christened C1
+ in [57], is a de-ductive extension of C1. Requiring less assumptions in order to obtain consistency of a complex formula in terms of the consistency of its components, Cilo (or even Cio, already, without recourse to the axiom (cl)) gives us some interesting results such as:
THEOREM 3.67 [ΓCio◦A] whenever [ΓCio◦B], for some subformula B of A.
Proof: Immediate, using (co1)–(co3).
as (A→⊥). But such an ‘informal’ bottom particle is simply no formula of our language! (And if it were, then we would have been done: all paraconsistent logics would turn out to be LFIs, in one way or another). This idea of Priest has already been criticized in Batens’s [13].
FACT 3.68 Cio makes some new additions to FACT 3.64 and to the rules displayed